Chapter 13. Case Study: Directed Graphs and Knots
Imagine that it’s 2 a.m. during finals week, and you’re scrambling to finish a research paper on topica obscura. Your adrenaline jumps when you find a relevant Wikipedia article, with links to more Wikipedia articles! You start clicking away, jumping from page to page in search of facts. An hour later, you realize you’re still clicking, but these are pages you’ve already visited. No matter which link you click, you can’t seem to discover any new pages!
If this has happened to you, you’ve unknowingly (and unfortunately) stumbled upon a knot in Wikipedia. Knots are a unique property of directed graphs. To understand them, it is necessary to have a basic understanding of directed graphs.
The graphs in the rest of this book are undirected. In an undirected graph, an edge represents a symmetric relationship between vertices. This abstraction is appropriate for some applications (such as acquaintance networks and transportation networks), but other applications involve asymmetric relationships (for example, links between pages in the World Wide Web).
Wikipedia links are also asymmetric. Page A might link to page B, but page B doesn’t have to include any links to page A. To fully describe the relationship between pages A and B, we need two bits of information: whether A links to B, and whether B links to A. This is the essence of a directed graph. In a directed graph, a directed edge, , is an edge ...